Questions tagged [philosophy-of-mathematics]

Philosophy of mathematics asks questions about mathematical theories and practices. It can include questions about the nature or reality of numbers, the ground and limits of formal systems and the nature of the different mathematical disciplines.

782 questions
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What are the main issues on which the schools of Intuitionism, Formalism, and Logicism disagree?

What is the difference between Intuitionism, Formalism, and Logicism? Namely - on which issues do they disagree? And what is the relation of those schools of thought to Platonism, Nominalism, and ...
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Are there recent coherence theory of truth for mathematical truths?

Are there any recent works (papers, books, etc) in philosophy of mathematics where it is given an account of mathematical truth in terms of a coherence theory of mathematical truth? I am interested ...
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Size of infinite sets [duplicate]

Cantor's method of comparing set size uses one to one correspondence i.e. existence of a bijection. Now, set A = (0,1) and set B = (0,2). Using function x -> 2x, every element of set A can be uniquely ...
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Is there an alternative to Cantor's cardinalities that makes proper subsets smaller than their sets?

Cantor defined an infinite set as a set whose subset can be placed in a one-to-one correspondence with its subset. That is, take the set of all natural numbers: {0, 1, 2, 3, 4,...}. From that set, you ...
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Ethics and Mathematics [on hold]

I am laying out a relationship between mathematics and ethics which I'm not sure is correct. In mathematics, we have axioms, statements we assume to be true. All the rest is logical deductions from ...
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Examples of theories that assume the existence of an “External Reality”?

In this paper written by physicist Max Tegmark (https://arxiv.org/pdf/0704.0646.pdf) it talks about "External Reality Hypothesis". Specifically, he says: Although many physicists subscribe to the ...
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Does Tegmark himself include paraconsistent mathematical structures in his mathematical multiverse hypothesis?

Tegmark postulates in his hypothesis that all possible mathematical structures would exist. But does he include also possible mathematical structures described by other types of logic like ...
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How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert quotes from Wittgenstein's "notorious paragraph" (§8 of Remarks on the Foundations of Mathematics, Appendix 3) in http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6....
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Mathematics and disagreements

I was just pondering as a mathematics major, is there a particular instance where a mathematician's work doe NOT require agreements among peer scholars of mathematics to determine the quality of the ...
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Help wanted - need descriptor for a partcular type/form of argument

I am writing a paper on cognition, and to simplify my discussion I need an adjective or descriptor for particular category of argument as follows: I am arguing for the necessity of a construct with a ...
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What should philosophers know about math and natural sciences?

My question is whether a lack of knowledge about formal mathematics or theoretical science in general would have an impact on a philosopher's ability to think and make judgments. Why should a ...
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Mathematical Universe

I have a beginner question on the type of claims this book or similar theories make. That book claims that universe is math structure. I just want to clarify if I correctly understood his goal: Does ...
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Can zero be defined without some definition of one? Can one be defined without some definition of zero?

I would prefer to ask this in the math community, but that crowd is hostile toward anything hinting of philosophy. It is my contention that a construction of the real number system which begins with ...
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Can a different universe be built with three dimensions? [closed]

Can you theoretically create a universe that will have the same dimensions but will look different? For example, a paper that has a limit, or another similar dimension can it be different?
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The Axiom of Choice (AoC) in set theory famously gives rise to controversial and counterintuitive theorems. (Examples: Banach-Tarski paradox and existence of non-measurable sets.) I'm aware of some ...
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What is the relevance of applicability to the natural sciences in pure mathematics?

I think I am coming to a good, new understanding of the relationship of pure mathematics to the natural sciences. A major concern of mine is just how reliable is rigorous (characteristically "pure") ...
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Does Tegmark's hypothesis include dynamical mathematical structures?

Tegmark's hypothesis is the idea that mathematical structures are physical and thus have physical existence (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) Zuse's thesis says that ...
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Relation of Mathematical Propositions to Natural Language

Treating Natural Language as a language game, what role does it play in our understanding of mathematics? Does natural language provide meaning to mathematics? Does a proof of a conjecture, say FLT,...
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Is there a natural example of a non-self-referential semantic paradox in philosophy?

A commonly studied paradox is the liar's paradox. The liar's paradox is to determine whether "this statement is false". The usual resolution is to state this the sentence is not actually a statement ...
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Can infinity be made finite in certain conditions?

In mathematics there are not only infinitely big numbers, but also infinitely small numbers. One can consider arbitrarily small numbers that can exist only in the mathematical world. For example, ten ...
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what is the ontology-ideology distinction in phil of math

Quine proposed a distinction between ontology - the doctrine of what there is - and ideology - the complex terms and predicates expressible in one's theoretical language (though in his 1971 paper he ...
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Why is 2 + 2 = 4? [closed]

It is clear that 2 + 2 = 4. It is also clear that applying the successor function on 1 yields the next number, i.e. 2, and this operation can be repeated infinitely. This method can be used to verify ...
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Can a problem be solved if there exists no solution for it in any context?

Is finding a solution for a problem in a given context is an attempt to find a solution for the problem in another context? For example, it seems some of the hardest problems of real analysis were ...
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Philosophy - If Space and Time are infinite and therefore infinite copies of us would end up existing, then wouldn't we still be gone after we die?

I have been pondering a question in my head. If Space and Time are infinite, then does that mean that Nietzsche's Eternal Return theory is true in the way that my life would recur, that when 'I' ('I' ...
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Syntactic VS Semantic Provability

Consider a new Conjecture C. The task is to determine whether the conjecture is true or false. Now let us suppose, after working very hard, we are finally able to establish the truth or falsehood of C....
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If we assume logic is correct, does it imply that our consciousness proccesses real information?

[major edits] Even if our consciousness is an illusion (even in the sense Denett suggests), the mere fact we see some information flowing across the universe means there is at least something that ...
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What's the meaning of this quote of Pythagoras on the good and bad principle?

Simone de Beauvoir attributed the following quote on the good and bad principles to Pythagoras in The Second Sex, page 114 : There is a good principle that created order, light, and man and a bad ...
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Correct Way of Handling A Corollary of A Corollary?

I have a conclusion S that is moderately interesting. While the corollary of S is more interesting, the corollary of the corollary of S is extremely interesting. Should I just label them corollary 1 ...
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Is getting 100 Heads in a row from a fair coin a miracle or not?

Suppose a man continues to toss a coin until he gets 100 heads in a row. Suppose the outcomes of all tosses from the 9999901th toss to the 10 millionth toss are all heads and 100 heads in a row didn't ...
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Probability calculus and Quantum Mechanics [closed]

I am not an expert and probably this question highlights this. Anyway, is the probability calculus used in Quantum Mechanics? Does the concept of probability adopted in Quantum Mechanics satisfy the ...
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Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?

So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a ...
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Discovery VS Invention in Mathematics [duplicate]

Asking specifically in the context of philosophy of Mathematics, on what basis do we classify or should classify a new expression as a Creation of Mind (Invention) OR a Discovery?
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Philosophy - Does Einstein's Block Universe theory prove Nietzsche's Eternal Return theory is true?

If the Past, Present, and Future all exist in exactly the same way, then every single moment would be a ‘Now’ moment for me. it would also mean that me being dead in the future is equally real in the ...
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Could generalization of scientific theories be possible by just adding an ad hoc hypothesis?

In a seventeenth century world the Newtonian model did mostly very well to describe how gravity works in the universe and did well with most empirical evidence of that time. Of course now we know that ...
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Is a proof still valid if only the author understands it?

Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to ...
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For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?

Mathematical realists believe that mathematical entities exit independently of human minds. Mathematical objects have an objective independent existence, and they are discovered by mathematicians, not ...
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Why is there something instead of nothing?

A simple but fundamental question. The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself debated. It ...
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Was Kant an Intuitionist about mathematical objects?

In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-...
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What are the discoveries that have been possible with the rejection of positivism?

I am wondering if the rejection of the positivism movement in philosophy lead to any major discoveries in mathematics and natural sciences? I am thinking it might have been able to contribute to those ...
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What does this Jacques Hadamard quote mean?

What does this Jacques Hadamard quote mean? The shortest path between two truths in the real domain passes through the complex domain. Is this a philosophical statement? what is its mathematical ...
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Is there any physics-model version of Tegmark's hypothesis?

Tegmark's mathematical universe hypothesis is very interesting (https://en.m.wikipedia.org/wiki/Mathematical_universe_hypothesis) but it has virtually no support among physicists because it is too ...
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The nature of Nominalist Formalism

In this entry in the stanford encyclopedia of philosophy, it is stated that the theory of nominalist formalism deals with the metatheory problem of formalism as follows: Commendably, Goodman and ...
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People who study mathematical logic make arguments about logic itself. So it seems that people take for granted an "intuitive logic" (otherwise, how would they form arguments?). So the observation is ...
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Mathematical Consensus

Can anyone give me a reason why mathematics may require consensus to determine the quality of knowledge from the general mathematical community? Also what would be the counter to such a claim, as in ...
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Are mathematical results influenced by the way we reason?

Intuitions of mathematicians, and the mathematics they develop, are ostensibly influenced by whether they primarily rely on visual_spatial and/or verbal_symbolic reasoning skills. Is it fair to say ...
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Where can I learn about the philosophy behind mathematical and logical proofs?

I'm looking for something that dives into the philosophical idea of a "proof," and explains how the subjects of mathematics and logic deal with it. Does anyone have any book or article recommendations ...
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Does philosophy of mathematics affect mathematical research?

I am interested in a special case of the general question about whether the philosophy of X has an effect on the research or practice of X. My special interest is in the area of mathematics. I am a ...
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This may be a duplicate question. What is the best way to get started with the philosophy of mathematics? Given that I know (from university) the basics that are discussed (Set theory, Russell's ...